The former chief of deportations in the Bush administration will testify to Congress on Tuesday that President Obama’s new nondeportation policies would have let the Sept. 11 hijackers remain in the country even if they had been picked up in the months before their deadly attacks.

And the current chief of the union that represents Immigration and Customs Enforcement agents will tell the House Judiciary Committee that ICE agents are now required to wait until most illegal immigrants have three misdemeanor convictions before they can be arrested and put in deportation proceedings.

“Most Americans would be surprised to know that immigration agents are regularly prohibited from enforcing the two most fundamental sections of United States immigration law,” said Chris Crane, president of National Immigration and Customs Enforcement Council. “According to ICE policy, in most cases, immigration agents can no longer arrest persons solely for entering the United States illegally.”

As momentum builds in the Democrat-controlled Senate to pass a broad bill legalizing illegal immigrants, House Republicans kick off their side of the debate Tuesday with a two-part hearing looking at the need for legal immigration and reviewing Mr. Obama’s record on enforcement and border security.

Mr. Obama’s first four years were characterized by record deportations of aliens with criminal records, but fewer rank-and-file illegal immigrants being deported.

His administration has issued several new policies that direct agents to focus on those with serious criminal histories, and it has created a new program for illegal immigrant young adults brought here by their parents to give them official legal status.

His moves have won support from immigrant-rights advocates, who say he’s blunted the worst abuses of a broken immigration system, but have earned derision from those who want to see a crackdown on illegal immigration.

The Judiciary Committee will hear from eight witnesses, including San Antonio Mayor Julian Castro, who delivered the keynote address at Democratic National Convention last year, and former ICE Director Julie Myers Wood, who ran the agency for the last few years of the Bush administration.

Mr. Castro will tell the committee that there is bipartisan momentum for getting a bill done, and will defend the Obama administration’s record.

“In Texas, we know firsthand that this administration has put more boots on the ground along the border than at any time in our history, which has led to unprecedented success in removing dangerous individuals with criminal records,” he said in his prepared testimony.

But Ms. Wood will tell the committee the focus on criminals “poses a potentially serious threat to our system.”

“It sends a message to those that seek to cause harm: if they can come in the United States illegally, but not immediately commit any additional crimes, they are likely to be left alone. Left alone to plan, take steps, cause harm,” she said.

She said the administration policy would have overlooked “individuals like several of the 9/11 hijackers, who ‘merely’ lied to obtain state identification documents or on their visa applications.”

The House hearing comes as the Republican Party is struggling with how to handle immigration.

Four Republican senators are working with four Democratic senators on legislation to legalize illegal immigrants. But House Republicans have been less open about how far they are willing to go.

“Before we rush to judgment, we need to carefully look at the current laws on the books to see what is and isn’t working,” said Rep. Robert W. Goodlatte, Virginia Republican and the new chairman of the Judiciary Committee. “Reforming our nation’s immigration laws is a massive undertaking and is too important to not examine each piece in detail.”

“Will this be the same old anti-immigrant wine in a new bottle? Or is the GOP finally ready to start fresh on immigration and work to rebuild its relationship with Latinos and other growing parts of the electorate?” he said.